Continuity of minimizers to weighted least gradient problems
| Autor: | Andres Zuniga |
|---|---|
| Přispěvatelé: | Indiana University [Bloomington], Indiana University System, Department of Mathematics, Indiana University, Indiana University System-Indiana University System |
| Rok vydání: | 2019 |
| Předmět: |
Mathematics - Differential Geometry
Weight function Boundary (topology) [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] 01 natural sciences Domain (mathematical analysis) Combinatorics symbols.namesake Mathematics - Analysis of PDEs Maximum principle FOS: Mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] 0101 mathematics ComputingMilieux_MISCELLANEOUS Mathematics Minimal surface Applied Mathematics 010102 general mathematics Zero (complex analysis) Primary: 49Q20 Secondary: 49J52 49Q10 49Q15 010101 applied mathematics Differential Geometry (math.DG) [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] Bounded function Dirichlet boundary condition symbols Analysis Analysis of PDEs (math.AP) |
| Zdroj: | Advances in Nonlinear Analysis Advances in Nonlinear Analysis, De Gruyter, 2019, 178, pp.86-109. ⟨10.1016/j.na.2018.07.011⟩ |
| ISSN: | 0362-546X 2191-950X |
| DOI: | 10.1016/j.na.2018.07.011 |
| Popis: | We revisit the question of existence and regularity of minimizers to weighted least gradient problems on a fixed bounded domain, subject to a Dirichlet boundary condition, in the case where the boundary data is continuous and the weight function is C^2 and bounded away from zero. Under suitable geometric conditions on the domain in R^n we construct continuous solutions of the above variational problem in any dimension n>=2, by extending the Sternberg-Williams-Ziemer technique to this setting of inhomogeneous variations. We show that the level sets of the constructed minimizer are minimal surfaces in a conformal metric determined by the weight function. This results complements the approach of Jerrard, Moradifam and Nachman since it provides a continuous solution even in high dimensions where the possibility exists for level sets to develop singularities. The proof relies on an application of a strict maximum principle for sets with area-minimizing boundary established by Leon Simon. 27 pages, 1 figure. Several improvements were made following the referee's suggestions. To appear in Nonlinear Analysis |
| Databáze: | OpenAIRE |
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